Detailed study of the decay of ³²Ar

The first step in the data analysis is the determination of the $\gamma$-ray energies and their branching ratios. Figure 2 shows the spectrum obtained from the eleven germanium segments used in the analysis. The peaks labelled with letters belong to the decay of ³²Ar. Labels $a-e,j,l,p$, are from levels in ³²Cl, in the $\beta$p daughter ³¹S (peak $f$) or from the decay of the $\beta\gamma$ daughter nucleus ³²Cl ($g-i,k,n-o$). Peak $a$ is not visible in the $\gamma$-ray singles spectrum. It becomes visible only in the $\gamma-\gamma$ coincidence spectrum. In addition, it is partially cut by the ADC thresholds for some of the crystals. Therefore, it is difficult to determine its exact energy and branching ratio. We refrain from doing this, as its branching ratio is the same as the one of the $\gamma$ ray at 1078.6 keV (see decay scheme in figure 6). Peak $f$ is visible in the singles spectrum, but much better evidenced in a proton-$\gamma$-ray coincidence spectrum, as shown in the inset of the upper part of figure 2. The other insets show expanded views of the singles spectrum to better visualise the $\gamma$ rays of interest.

Table 2 summarizes the $\gamma$-ray data from this work and from Björnstad et al. bjornstad85 for states in the $\beta$-decay daughter ³²Cl populated by a Gamow-Teller $\beta$ decay and for the $\gamma$ ray in the $\beta$p daughter ³¹S. Reasonable agreement is obtained for all energies. As the $\gamma$ lines from the decay of ³²Ar were not included in the energy calibration, we can build the averages from the present work and from Björnstad et al. For the branching ratios, good agreement is also obtained, except for the $\gamma$ ray at 1249 keV. The value from our work is overestimated, because this $\gamma$ ray is always in coincidence with protons, for which the trigger probability of the data acquisition is significantly higher than in the case of $\beta$-delayed $\gamma$ rays triggered by the $\beta$ particles in the DSSSDs. We refrain therefore from averaging the two values.

The absolute branching ratios can be obtained from the complement of the total proton emission branching ratio from Bhattacharya et al. of BR${}_{p}(tot)$ = 35.58(22)%. We assume here that the ground-state feeding is negligible (see below). The results of this procedure are given in the last row of table 2.

Three $\gamma$ rays were observed in the work of Bhattacharya et al. to de-excite the IAS of the ground state of ³²Ar in ³²Cl. These are peaks $j$, $l$, and $p$, where we admit that peak $p$ would have been difficult to identify without knowledge from the work of Bhattacharya et al. Nevertheless, the energies and branching ratios determined show good agreement with the results of Bhattacharya et al. The energies as well as the absolute branching ratios and their averages are presented in table 3 and will be discussed below.

Gamma rays were not only observed in the $\beta$p daughter nucleus ³¹S, but also in the decay of the $\beta\gamma$ daughter nucleus ³²Cl. These are given in table 4. To obtain their absolute branching ratios, we normalised them with the total $\beta\gamma$ branching ratio calculated from the total proton branching ratio as given by Bhattacharya et al. bhattacharya08 . In the table, we also present a comparison to data from the literature nndcA=32 . Good agreement is obtained indicative of the fact that the extrapolated $\gamma$-ray detection efficiencies work well up to energies as high as 5 MeV.

Figure 3 shows the singles proton spectrum from DSSSDs 1, 3 and 6. A spectrum was also created with the same proton spectrum, however, in coincidence with a $\gamma$ ray at 1249 keV, the $\gamma$ ray de-exciting the first excited state of ³¹S to its ground state. The result is also shown in figure 3. A total of 30 proton lines were observed in the present work. A very weak peak at 912 keV observed by Bhattacharya et al. bhattacharya08 was not seen in the present work nor in the work of Björnstad et al. bjornstad85 .

All proton lines were fit by a Gaussian and a low-energy tail to take into account the possibility that the full charge was not always collected in the silicon detectors or that interference effects create a low-energy tail. One single Gaussian plus a tail was sufficient for all proton groups to get a good description of the experimental peak, except for the line at 3353  keV, which is due to the emission of a proton from the IAS, the strongest peak in the spectrum of figure 3. In this case (peak 14), a second Gaussian was added. The integral of the Gaussians yielded the number of counts in the different peaks and allowed the determination of relative branching ratios. In table 5, we present the energy values of these proton lines and their relative branching ratios normalised to 1000 for the strongest proton line from the IAS. The data from the present work are compared to the data available from the work of Björnstad et al. bjornstad85 and Bhattacharya et al. bhattacharya08 , this latter work giving only proton lines with energies below about 4 MeV if they feed the first excited state in ³¹S (except for peak 1), and all proton lines above 4 MeV.

In the following, we quickly discuss all proton lines from table 5 individually. For this discussion, we will use the average proton energies. The numbers correspond to the peak numbers used in figure 3 and in table 5.

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  1: In the present work, we can see the 608 keV peak only under the condition that we require a signal in two different strips (one being from the proton, one from the associated $\beta$ particle) and keep only the higher energy. This condition drastically reduces the $\beta$ background at low energies and makes this peak visible. For a correct normalisation for this peak, the 3353 keV line was also analysed under the same condition. It is a proton to the ground state of ³¹S.

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  2: The 912 keV line is not identified in our data. Only Bhattacharya et al. have a tiny proton branch at this energy to the first excited state.

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  3: The 1211 keV line is seen in the three experiments as a proton emission to the first excited state in ³¹S with comparable intensities.

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  4: The authors from previous work did not identify a proton line at 1677 keV, but we have clear evidence for such a line in the singles and in the coincidence spectra. Björnstad et al. have also a small peak at this energy, however, they do not mention it. We attribute it to the decay to the first excited state of ³¹S.

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  5-6: Björnstad et al. identify a single peak at 2124 keV. However, in our data we have clear evidence for a ground-state decay at 2122 keV and a decay to the first excited state at 2146 keV. This is in agreement with the work of Bhattacharya et al. who observe also a peak to the first excited state at 2146 keV. The branching ratio to the ground state and to the first excited state as determined in the present work add up to the branching ratio given by Björnstad et al. So we determine from the average of our work and the one of Bhattacharya et al. the relative branching ratios to the ground and first excited states and use these value to distribute the branching ratio found by Björnstad et al. to the ground and first excited states for these authors.

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  7: Björnstad et al. have no peak at 2395 keV. In the present work and in Bhattacharya et al., this peak is visible as a decay to the first excited state, although with almost a factor of 2 difference in branching ratio.

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  8: The peak at 2423 keV is observed in the present work and by Björnstad et al. as a decay to the ground state in ³¹S.

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  9: The 2512 keV line is observed in the three experiments with comparable intensity. As Bhattacharya et al., we see it as a decay to the first excited state. Björnstad et al. identify it as a decay to the ground state. Due to the clear evidence from the present work and from Bhattacharya et al., we attribute it to the decay to the first excited state.

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  10: No 2616 keV peak is observed in the work of Bhattacharya et al. and of Björnstad et al., but there is a visible shoulder in the present data and in those of Björnstad et al., which we identify as a proton emission to the first excited state. The observation of this peak may be questionned due to the fact that there is no corresponding decay to the ground state of ³¹S. This needs to be investigated in a new experiment.

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  11-13: At 2775 and 3117 keV, Björnstad et al. have two peaks as in the present work, but these authors do not give them in coincidence with the $\gamma$ ray at 1249 keV, which could be due to the small branching ratio and their small $\gamma$-ray detection efficiency. We clearly see this coincidence and attribute these two peaks to decays to the first excited state. However, in our data, the 3117 keV peak is also in the ground-state spectrum with a branching ratio of 1.32% (peak 12). Bhattacharya et al. see in this region a broad structure at 2870 keV with a much higher relative intensity (30(10)%) as compared to 5.9(7)% for the present work and 8.3(10)% for Björnstad et al. The data of Bhattacharya et al. for the 2870 keV peak were not used for the averaging of the proton energies nor for the branching ratios. Like for peaks 5 and 6, we distribute the branching ratio of Björnstad et al. for the 3117 keV line to the ground-state and excited-state decay.

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  14: The 3353 keV peak is the most prominent peak for all three data sets. It is therefore taken for the normalisation. As will be discussed below, a small contribution in coincidence with the $\gamma$ ray de-exciting the first excited state in ³¹S cannot be completely excluded.

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  15-16: Around 3600 keV, we see a somewhat broader peak with one part decaying to the ground state and one to the first excited state of ³¹S. We attribute the 3583 keV line to the decay to the first excited state and the 3605 keV line to the decay to the ground state. Bhattacharya et al. see the decay to the first excited state, whereas Björnstad et al. see one peak ”predominantly” decaying to the first excited state, indicative that in their data there might be also decay strength to the ground state. We therefore distribute their decay branching ratio to both decays according to the relative branching ratios from our data and from Bhattacharya et al.

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  17: Nice agreement is obtained for all three data sets for the peak at 3649 keV as a decay to the first excited state.

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  18-19: The peak at 3732 keV in Björnstad et al. has at least two components. We find a ground-state peak at 3727 keV and an excited-state peak at 3782 keV, in agreement with Bhattacharya et al. If we sum the two peaks observed here for decays to the ground and first excited states, we come close to the integral of Björnstad et al.. We distribute their decay branching ratio to both decays according to the relative branching ratios from our data and from Bhattacharya et al.

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  20: The 3991 keV peak is observed in all three experiments with comparable intensities as a decay to the ground state.

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  21-22: For the peak at about 4340 keV, we clearly see it for the ground state decay (22) and in the coincidence spectrum (21). So we share its activity between the ground state and the first excited state also for Bhattacharya et al. and Björnstad et al.

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  23: The 4526 keV peak is seen in the three experiments, but Björnstad et al. claim to observe it predominantly in coincidence with the 1249 keV $\gamma$ ray, whereas Bhattacharya et al. see it mainly as a ground-state decay. We can only observe a ground-state branch at this energy. We attribute the strong ground-state decay observed by Bhattacharya et al. (0.54(5)% with respect to the IAS decay) to this peak 23. We do the same for the peak of Björnstad et al. (0.12(1)% absolute branching ratio). We neglect the small possible excited-state contribution ($<$0.03% with respect to the IAS peak 14) from Bhattacharya et al.

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  24-25: A proton group at 4625 keV is seen in the three experiments with comparable intensities as a decay to the first excited state. The 4867 keV peak decays also to the first excited state.

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  26-31: The proton groups at 4978, 5560, 5684, 5823, 6071, and 6365 keV are identified as decays to the ground state in ³¹S in all three experiments.

The last three peaks were evaluated with DSSSD 5. We found that the branching ratio determined with detectors 1, 3, and 6 was the same as with detector 5 within the error bars for proton group 29, but factors of 2 and 6 lower for the high-energy proton groups 30 and 31, respectively.

We note that the proton peak energies of Bhattacharya et al. are systematically higher than those given for our data and those from Björnstad et al. above about 4500 keV. Schardt and Riisager schardt93 mention that the energies of Björnstad et al. should be corrected, because these authors used an $\alpha$-particle energy calibration at high energies. However, the correction proposed moves the proton energies of Björnstad et al. even above the values of Bhattacharya et al. and far away from our proton energies. For the present paper, we keep the energy values from Björnstad et al.

The doublets that we observe at around 3130 keV, 3595 keV, and 4340 keV have been given particular attention. We first fit a single proton peak in the $\gamma$-coincidence spectrum and in the ground-state decay spectrum, this latter being obtained by subtracting the spectrum for the decay to the first excited state with the correct normalisation from the proton singles spectrum. This procedure was verified by fitting a doublet to the proton singles spectrum by keeping one of the two energies and the width of the peaks fixed. It allowed us also to determine the relative intensities of the ground-state and excited-state decays and to observe that none of the two alone gives the intensity of the singles-spectrum peak, when fitted with only one proton line.

The relative branching ratios were averaged (with the restrictions mentioned above) as given in table 5. With the technique used in the present work, we cannot determine the number of implanted ³²Ar in any precise way to transform the relative branching ratios into absolute proton emission branching ratios. In order to determine absolute branching ratios, we use again the total proton emission branching ratio of Bhattacharya et al. However, if we just add the integrals of the identified peaks to determine the total number of protons detected in the present work, the procedure is not correct. Some activity is not identified in peaks (e.g. activity between peaks 26 and 27). We therefore determined the total number of protons by fitting an exponential to the low-energy $\beta$-particle contribution in the upper spectrum of figure 3 and subtracted this contribution before determining the total number of counts in the spectrum, which corresponds to the total number of protons detected. If we compare this number to the sum of the integrals of all identified proton groups, we find that a small correction factor of 1.033(14) is needed. With this correction and the total proton emission branching ratio from Bhattacharya et al. of BR${}_{p}(tot)$ = 35.58(22)%, we can determine the absolute branching ratios given in the table.

From the number of protons in the singles spectrum after the subtraction of the $\beta$-particle contribution at low energies and from the number of protons in the spectrum conditioned by a $\gamma$ ray at 1249 keV after subtraction of random coincidences (see below), we can determine the proton branching ratio to the first excited state of ³¹S to be 2.5(3)%, which can be compared to the value obtained by Bhattacharya et al. of 2.3(4)%. Björnstad et al. found a value of 1.9(2)%, whereas we determine 2.1(2)% with a slightly different absolute normalisation for the latter data as already discussed by Bhattacharya et al. (see table 2).

Up to now, we have treated the proton and the $\gamma$-ray data almost independently. Only to better evidence certain proton groups most likely being due to a decay to the first excited state in ³¹S, we used proton-$\gamma$-ray coincidences to produce the lower-statistics spectrum shown in figure 3. However, the fact that a peak shows up in this $\gamma$-ray coincidence spectrum does not necessarily mean that the protons are really in coincidence with the $\gamma$ ray at 1249 keV, although, as we will show, this is the case most of the time. We note also that in our experiment, due to the total number of decays observed and due to our efficiencies, the detection limit for proton-$\gamma$ coincidences with one count corresponds to a branching ratio of 0.0015%. However, for higher-intensity proton peaks the accidental coincidence probability is much higher.

In order to evidence proton-$\gamma$-ray coincidences, we overlay in figure 4 the proton singles spectrum from DSSSD 1, 3, and 6 as well as from DSSSD 5 with a normalisation factor of 3 with the proton spectrum generated from the same detectors requiring a coincidence with a $\gamma$ ray between 1240 keV and 1254 keV (we do not subtract for the moment random coincidence, see below). However, as the $\gamma$-ray detection efficiency decreases the number of counts observed in this coincidence spectrum, we multiplied the number of counts in each channel by a factor of 35.1(19) determined as the average of the inverse of the $\gamma$-ray detection efficiency at 1249 keV and the ratios of the number of counts of the 1211 keV and the 2512 keV proton peaks in singles and with the $\gamma$-ray coincidence. As just multiplying this spectrum with a factor of 35.1 would yield a spectrum difficult to ”read”, we ”randomised” the multiplied spectrum by folding each event with a Gaussian with a sigma of 2 keV to smoothen the spectrum. The result is shown in figure 4 together with the proton singles spectra and the original $\gamma$-ray coincidence proton spectrum.

In these spectra, one clearly sees that for some proton groups one recovers almost the full proton singles intensity with the scaled coincidence spectrum (e.g. protons at 1211 keV, 2512 keV, 4867 keV). However, there is also significant proton activity under the peak at 3353 keV, which is supposed to be the decay of the IAS in ³²Cl to the ground state of ³¹S. So this peak should not be in coincidence with $\gamma$ rays at 1249 keV.

In order to determine the ”real” coincidence counts and remove random coincidences, we show in figure 5 the same $\gamma$-ray coincidence spectrum with the 1249 keV $\gamma$ ray and overlay a spectrum generated with a coincidence left and right of the 1249 keV peak. This latter spectrum should consist basically only of random coincidences. In addition, we determine the ratios between the number of counts in both spectra for 21 proton-peak regions, which include 25 of our 31 proton groups. For the analysis of the ratios in figure 5, we centered the regions, given by the vertical lines, at the energies of the proton groups (mainly for decays to the first excited state) with a width of $\pm$40 keV in the low-energy region and $\pm$50 keV for higher proton energies (the last 4 regions). Peaks 9-10, 12-13, 15-16, and 21-22 were evaluated within a single region. For some peaks, the intervals are smaller than $\pm$ 40 keV because two neighbouring peaks are too close in energy.

This evidences that we deal with almost the same number of counts e.g. for the region of the 3353 keV peak in both spectra. For regions with real coincidences (e.g. the peaks at 1211 keV, 2146 keV, 2775 keV, etc.), this ratio is small. However, often ground-state decays and excited-state proton groups overlay or are very close. A good ratio limit seems to be a value of 0.2 (see below), where real coincidence peaks have a smaller and random coincidence peaks have a larger value. From these ratios, we determine that peaks with numbers 3, 4, 6, 7, 9, 10, 11, 13, 15, 17, 19, 21, 24, and 25 are proton emissions to the first excited state of ³¹S. For peak 2, the situation is more difficult to analyse due to the presence of a strong $\beta$ background. According to Bhattacharya et al., it is a rather weak proton emission to the first excited state, which is not observed neither in our work nor by Björnstad et al. We keep it as a proton emission to the first excited state of ³¹S.

Another finding that might deserve particular attention in a future high-resolution study is the fact that, for the 3353 keV peak number 14, the random coincidences are exclusively located in the high-energy part of this peak. If one subtracts the random coincidences, a peak in the low-energy tail of this strong peak remains, which could be indicative of a small proton peak with a real coincidence with the 1249 keV $\gamma$ ray. This might be important because it reduces slightly the Fermi strength discussed below.

The ratio of 0.2 used above might seem somewhat arbitrary, but it is meant to be an indication only for the ratio of counts in the different regions between the two spectra. One can also subtract the spectrum with the random coincidence (condition left and right of the 1249 keV peak) from the coincidence spectrum (condition on the 1249 keV peak). Unfortunately, the statistics becomes rather limited. However, if we fit this difference spectrum, which should contain only real coincidences, with energies fixed to the values of table 5, the numbers of counts for peaks without real coincidences (i.e. peaks 5 and 8) are compatible with zero and confirm thus the above conclusions. As stated in the previous paragraph, counts remain in the low-energy part of peak 14.

In the preceding sections, we have been able to attribute all proton groups to decays from states in ³²Cl to two states in ³¹S, the ground state and the first excited state at 1249 keV. No decay to other states of ³¹S could be evidenced. Proton energies having a center of mass difference in agreement with a $\gamma$-ray energy of 1249 keV (we required a difference of less than 1.5 times the sum of the uncertainties of the proton energies) stem from the same level in ³²Cl, but decay to the ground and first excited states of ³¹S. Table 6 summarizes this information and gives the excitation energies of states populated in the decay of ³²Ar (calculated with a proton separation energy of $S_{p}$ = 1581.1(5) keV ame2016 ), the branching ratio to these levels and the center-of-mass energies of protons involved in the decay of these levels to the ground or first excited state in ³¹S.

If we compare the levels deduced in the present work with the adopted levels from the evaluation of states in ³²Cl nndcA=32 , we propose one additional level at 5530.6 keV, whereas for four states from nndcA=32 we cannot identify the corresponding proton peaks (excitation energies 2611 keV, 4167 keV, 4439 keV, 5794 keV). The first level comes from a ”private communication”. This level is observed in none of the $\beta$-decay experiments. However, it would correspond to a level clearly present in the mirror nucleus ³²P at 2740 keV. The second and third states come from the work of Björnstad et al. from two proton branches attributed to a ground-state decay. We clearly identify the two proton branches as decays to the first excited state of ³¹S. The last level comes from the very broad 2870 keV proton line identified by Bhattacharya et al. only and discarded in our analysis (see above). So from the four levels, we will keep only the 2611 keV line.

In addition to the unbound levels, which decay by proton emission, we give also the feeding of the bound 1⁺ level at 1168.5 keV of ³²Cl, which $\gamma$-decays to the ground state of ³²Cl. This decay takes place via a $\gamma$-ray cascade that can be determined with the $\gamma$ rays observed and attributed to the decay of levels of ³²Cl (see table 2).

All this information allows us finally to establish a rather complete decay scheme for ³²Ar given in figure 6. Information from the present paper is combined with the results from Björnstad et al. and Bhattacharya et al. and data from the literature e.g. for the half-life and the $Q$ value nndcA=32 . The branching ratios, the $\beta$-decay Q values and the half-life of ³²Ar allowed us also to determine the $log(ft)$ values for the different decay branches as given in the decay scheme.

The information gathered to establish the decay scheme can also be used to determine the Gamow-Teller strength distribution B(GT). However, this procedure is only correct, if all decay strength can be included. As stated above, about 3.3% of the total decay strength from unbound levels in ³²Cl could not be attributed to a proton peak. This induces certainly only a small error in the B(GT) distribution, in particular at high excitation energy. We will determine this strength distribution in the standard way by using the identified peaks and their branching ratios as well as in a more precise way by using directly the proton spectrum.

For this latter purpose, we use the proton singles spectrum (from detectors 1, 3, and 6 up to 5.70 MeV, from detector 5 above), subtract the $\gamma$-coincident spectrum from the same detectors (multiplied with the factor of 35.1) to obtain the spectrum of protons to the ground state of ³¹S, take the $\gamma$-coincident spectrum and shift it by 1249 keV, and add it back to the ground-state spectrum with the correct multiplication factor. This new spectrum can then be converted into a spectrum of population of excited states in ³²Cl. We next exclude the region of the IAS. The total number of counts in the spectrum obtained together with the total proton branching ratio allow us to give to each event and thus to each channel of this spectrum a branching ratio. The excitation energy, to which each channel corresponds, enables us to determine the $Q$ value for each channel and then the Gamow-Teller strength B(GT) as

with $K$ = 6143.6 s (see e.g. hardy20 ) and the axial-vector coupling constant $G_{A}$ = -1.2756 pdg20 . $f$ is the statistical rate function towner15 depending on the $\beta$-decay $Q$ value and $t$ is the partial half-life defined as

with BR being the branching ratio attributed to each spectrum channel. The B(GT) value for the feeding of the bound state at an excitation energy of 1169 keV is calculated in a similar way. We used T_1/2 = 98(2) ms nndcA=32 .

The result of this procedure is shown in figure 7 together with the B(GT) distribution determined with the identified proton peaks. One can see that starting from about 6 MeV excitation energy, the B(GT) distribution obtained from the identified proton peaks stays below the B(GT) distribution from the full proton spectrum.

We compare our results to calculations obtained in the sd-shell model space with protons and neutrons in the $(1s_{1/2},0d_{5/2},0d_{3/2})$ set of orbitals. We will use the so-called “universal” sd (USD) type Hamiltonians. Universal means that the same set of single-particle energies and two-body matrix elements can be applied to all states whose wavefunctions are thought to be dominated by sd-shell model configurations. The two-body matrix elements are allowed to have a smooth mass dependence. This is in contrast with the ab-initio in-medium similarity renormalization group (IMSRG) stroberg17 ; stroberg19 method where the Hamiltonian is nucleus dependent. At present, the rms deviation between the experimental and theoretical excitation energies for IMSRG is about 500 keV. Making this universal assumption, one can fine-tune the two-body matrix elements to minimize the deviation between experimental and theoretical energies and reduce the rms deviation down to about 150 keV.

We will use five USD-type Hamiltonians: (i) the original isospin-conserving USD Hamiltonian based on data for sd-shell nuclei with $N\geq Z$ available up to 1983 brown88 ; wildenthal84 , (ii) the improved isospin-conserving USDA and USDB Hamiltonians based on data for nuclei with $N\geq Z$ available up to about 2006 brown06 , and (iii) the most recent isospin-nonconserving USDC and USDI Hamiltonians based on data for all sd-shell nuclei magilligan20 . The isospin-nonconserving part contains the Coulomb interaction and an isotensor strong interaction.

The feeding of the IAS in ³²Cl can be determined in part from its decay by proton emission to the ground (proton peak 14) and the first excited (peak 6) states. Although the IAS is largely proton unbound and one would expect that it decays only by proton emission, Bhattacharya et al. have shown that it decays also by $\gamma$-ray emission to three states in the $\beta$-decay daughter nucleus ³²Cl. We can clearly identify the first two $\gamma$ rays (labelled $j$ and $l$ in table 3), whereas we consider our evidence for the highest energy $\gamma$ ray only tentative (labelled $p$ in the same table). This $\gamma$ decay of the unbound IAS is favoured due to the isospin-forbidden character of proton emission from the IAS. Only due to a most likely quite small T=1 impurity of the IAS, this proton emission can happen. The proton-emission branching ratio is determined in our work to be 20.77(17)%, whereas the $\gamma$-ray branches sum up to 1.86(18)% yielding a total branching ratio to the IAS of 22.63(25)%. All these values are in perfect agreement with those already given by Bhattacharya et al. bhattacharya08 . In fact, this is not very astonishing, as they are averages from the present data with those from Bhattacharya et al. and from Björnstad et al. and include in addition the total proton-emission branching ratio from Bhattacharya et al.

This super-allowed branching ratio allows us to determine the $ft$ value of the super-allowed decay branch to be 1514(35), where we used nndcA=32 a half-life of 98(2) ms, a $Q$ value of 11134.3(18) keV for the ground-state decay and thus a $Q$ value to the IAS of 6088.0(18) keV yielding f = 3494.6(58). An electron-capture probability of $P_{EC}$ = 0.067% firestone was also included. To determine the corrected $\mathcal{F}t$ value (see e.g. hardy20 ), we use in addition the theoretical isospin-breaking correction $\delta_{c}$ =2.0(4)% and the radiative correction $\delta_{R}$ = 1.145(41)% from Bhattacharya et al. and obtain $\mathcal{F}t$ = 1501(35) s. As this is a value for a T=2 transition, it can be compared to the value determined for the well-known 0${}^{+}\rightarrow$ 0⁺, T=1 decays compiled regularly by Hardy and Towner hardy20 divided by a factor of 2. The value from the latest compilation is 3072.24(57) s, which shows that the presently determined value is in agreement with expectations.

The isospin non-conservation that is contained in the USDC and USDI Hamiltonians results in isospin mixing between the 0⁺, T=2 state and 0⁺, T=1 states. The strongest mixing is with a state 226 (308) keV lower in energy with the USDC (USDI) hamiltonian. This results in a splitting of the total Fermi strength of B(F)=4 into B(F)=3.87 (B(F)=3.92) for the T=2 state and 0.13 (0.07) for the T=1 state below the IAS for the USDC (USDI) interaction. The remaining Fermi strength is distributed over the higher-lying 0⁺ states in the region of 6 to 10 MeV not observed in the present experiment.

The calculated proton-decay width of this 0⁺, T=1 state is 36 keV (C²S=0.042) with USDC and 37 keV (C²S= 0.042). A possible candidate for the 0⁺, T=1 state below the IAS could be the experimentally observed state at 4779 keV. However, the branching ratio to this state is about a factor of 25 lower than predicted. Another candidate could be the state at 4561 keV. However, its branching ratio is about a factor of 8 lower than predicted.

From the branching ratios for the IAS, we determine a ratio of the decay widths of R = $\Gamma_{p}$/$\Gamma_{g}$ = 11.2(11). The literature gives a total width of the IAS of $\Gamma$ = $\Gamma_{p}$ + $\Gamma_{g}$ = 20(5) eV nndcA=32 . These values allow us to determine $\Gamma_{g}$ = $\Gamma$/(1+R) = 1.8(5). This compares well with the calculated value of $\Gamma_{g}$ = 1.3 eV from both USDC and USDI. The experimental proton width is thus $\Gamma_{p}$ = 18.2(5) eV and has to be compared to the calculated values of 525 eV for USDC and 600 eV for USDI. Since the proton width $\Gamma_{p}$ comes from isospin mixing, the conclusion is that the isospin mixing from the experiment is much smaller than in the calculations, which will have consequences for the interpretation of the isobaric multiplet mass equation for A=32 signoracci11 . This conclusion is confirmed by the determination of the experimental B(F) value calculated from the $ft$ value of the super-allowed decay branch to be 3.991(93) in agreement with the expected value of 4, but also with the theoretical values of 3.87/3.92.